 Welcome back to our lecture series, Math 3130, Modern Geometries for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. So in previous lectures, we began our discussion talking about Hilbert axioms of Euclidean geometry. Hilbert put his axioms into several families of geometric axioms, and we're going to study those families individually stacking them on top of each other. So the first family of axioms that we talked about due to Hilbert were the so-called incidence axioms. So this was line determination, point existence, non-coloniality, and secancy. For which any geometry that satisfy those four axioms, we called it an incidence geometry. The next family of axioms are known as the betweenness axioms, or sometimes they're called the order axioms. The word order will make more sense in the future. We'll introduce the four betweenness axioms in this video. But remember how the axiomatic method works. The first step is we're going to look at undefined terms. We introduce terms that don't necessarily have definitions. All right, undefined terms. From there, we then introduce things like definitions. These are things that we then can define using these undefined terms. Axiom three, excuse me, step three is then rewrite the axioms. And then step four is we then do the theorems. For which, while we spend most of our time proving the theorems, it's important that we do all of these steps right here. So actually, before we introduce a new undefined term that has to do with the betweenness axioms, I actually want to introduce a definition that really is a definition of incidence geometry. This doesn't need any betweenness notion here, but we haven't really needed it up to this point. And it's similar to some definitions of betweenness terms that we're going to introduce in just a moment. So we're going to make this definition right now and then we'll go through our steps here. So again, this is a definition for incidence geometry here. So let A and B be two points inside of an incidence geometry. The line determination axiom guarantees there exists a unique line that contains A and B. We are going to denote that line as A, B with a line symbol on top of it. So you have this double arrow going on here. If you're trying to do this in latex, oh boy, what was the command for that? Something like backslash left, right arrow. I think it's something like that. If I'm wrong, I'll correct it in the comments below. But something like that should then give you that symbol in latex if you wanted to type that thing up. But the idea is this is then going to determine the, this is the symbol for the line that's uniquely determined by the points A and B. Okay, so we've just specifically saying the line that's determined by A and B. So instead of having to give all the lines names all the time, like this is L1, this is L2, this is L3, we can just refer to the line by the points determined by that we given by, that's given by line determination. All right, so that gave us one more definition for the incidence axioms. What about the between this axioms now? Well, what's the undefined term that we're gonna introduce now? Well, this is the idea of betweenness. What does it mean to be between points, right? So naively, we're trying to express a sort of a local behavior. We have three points and actually forms a trinary relationship between them. So if you have three points A, B and C that are distinct, we say that B is between A and B and we'll denote this as A-B-C. So this is, like I said, a trinary relationship we call betweenness. So B is between A and C. What does that mean? Well, it's an undefined term. We don't say what it means. We interpret what it means. We'll do that some other time, all right? We will provide axioms which essentially are definitions of the undefined terms because it does put criteria on what those undefined terms have to do. But right now we're working on, we have our undefined term. So that's what it means to be between. So B is between A and C. Now let's go on to some definitions. So betweenness naturally gives us a trinary relationship. So A-B-C. We can then can turn this into a four-ary relationship. So relationship on four points. So we can write A-B-C-D. What does that mean? It actually means four statements about betweenness. So if you have A-B-C-D that means A-B-C. So that is B is between A and C. It's gonna mean A-B-D. So B is between A and D, right? It's gonna mean A-C-D. So C is between A and D. And then likewise we're gonna have B-C-D which means that C is between B and D. Now when I say these things out loud, these things get a little bit hard to hear sometimes because letters B-C-D all rhyme with each other. So if I ever go to like military alphabets, so like Alpha, Bravo, Charlie, Delta, just be aware that the word like Charlie, Charlie starts with a C, so Charlie is C but it doesn't rhyme with anything else. D, Delta starts with a D, Bravo starts with a B. The idea of this phonetic alphabet, excuse me, military alphabet is just that when I say the word, the first sound is the letter I'm giving a name to, but it's to avoid confusion. Did you say B or D there? If I ever say Bravo or Delta, that's what's going on there. So if we have a four relationship of betweenness, it really means these three relationships altogether. So that's the definition of this symbol right here. And so using notions of betweenness, we can generalize the idea of the line determined by A and B. So that's why we delay this definition until just right now. We can introduce the idea of a line segment. So we have two points A and B which have a notion of betweenness defined, right? Then we can define the segment AB. So notice here, we have just a line on top with latex. This will just be overline, overline A and B, like so. This is defined to be the set of all points that are either A, B, or are between A and B, okay? So we take this segment here, we're gonna take all the points C that are between A and B and then we include A and B itself. So it's like an interval, right? We have the end points A and B but we also have everything between A and B. If we have a notion of betweenness, we can then define a line segment or just a segment for short, okay? Now be aware at this moment, we don't actually know that being between necessarily means you're co-linear. So is AB a subset of the line AB? We don't actually know that yet because we have no meaning of word between means. It's just another fine term. But since it's now a word in our vocabulary, we can describe things using betweenness. Related to the idea of a segment, we also have the idea of array. What is array? Array is gonna be defined to be the set where it includes the line segment AB but we also union with it everything that extends it. What do I mean by extension here? So we're gonna look for all point C such that B is between A and this new point C like so. And so this set called the array for which again, this will be in latex if you care about these things. This should be backslash right arrow AB, something like this. The direction does matter in this situation because while we get the symbol AB, which we'll find in a little bit that actually doesn't matter the direction. I'm getting a little ahead of myself here, of course. But are we going from A to B to C? Are we going from B to A to C? That could give you a different thing. So the range matter in that situation. All right, so one more bit of vocabulary before we then provide the betweenness axioms here. We define the idea of same side of a line. So we say that two points that are not on a line L are on the same side if the segment AB doesn't intersect the line. Because after all, a line, because of incidence geometry can be identified with the points on that line. The points on a line characterize the line in incidence geometry. And the segment AB is also a set of points. So what's the intersection of this? If the intersection's empty, if the intersection's empty, then we say AB are on the same side of the line. If the intersection is not empty, then we claim they're on opposite sides of the line. And the intuition behind this is the following. We have our line and we have two points. If A and B are on the same side of the line, then the line segment doesn't intersect. But if they're on different sides of the line, then the line segment does intersect it. So with this now taking care of, let's proceed to talk about the four betweenness axioms. So just like incidence, there's gonna be four betweenness axioms given by Hilbert. The first, and we're gonna give them names. These are not standard names. These are names created just by myself for the use of this lecture series as we talk about these things. Because if I give them names like B1, B2, B3, B4, we'll never remember what they are. But if we give them names that basically sound like what they are supposed to be, then we're in a situation where we can actually remember them, recall them, we can talk about them amongst ourselves. As I refer to them in future videos, you'll know exactly what I'm talking about. So the first axiom is called the collinearity and symmetrization axiom, which I'll typically call it collinearity for short. This of course should not be confused with the non-collinearity axiom for incidence geometry. That one said that not all points live on the same line. With regard to the betweenness axiom, collinearity means that if A dash B dash C, that is if B is between A and C, that means the points A, B, C are collinear. So this axiom actually guarantees that the betweenness relation implies the points are collinear. They're on the same line. Therefore the line determined by AB is equal to the line determined by AC, which is then determined by the line Bravo Charlie, like so. So those are all the same lines because they're collinear. So that's why the collinearity name comes into play there. The symmetrization just means that this trinar relationship is symmetric. So if A dash B dash C is true, it's also true that C dash B dash A is true as well. And so collinearity gives you that if you have a betweenness relationship, you have its reverse betweenness relation is also true, and these three points are collinear with each other. Okay? Axiom two, this is gonna be one of the most important axioms all. I mean, it's hard to classify who's the most important in situations. I mean, cause we use this one all the time, B one, but we just, it's so fluid that we don't even mention it most of the time. Extension is a very important axiom as well. The extension axiom tells us the following that if you have two points A and B that are on the same line, so these, they're on the same line here, which call it L, then extension gives us that there exists a third point C on the line L such that A dash B dash C holds. So you have B is between A and C. So the idea is if you have your line with the points A on it, B on it, there's always another point down the road that extends it. But if you apply an induction argument here, so by A and B we could extend it to get C. If you took A and C, you could extend that to get D. If you used another point to get E and you can keep on going by induction, the extension axiom can be used over and over and over again, which actually tells that if you have the extension axiom, your geometry actually must be infinite. If you have extension, then all lines contain at least countably many points by induction. And therefore, you have a non-finite geometry, aka it's an infinite geometry. So the finite geometries we explored before, which were really cool incidence geometries, we're gonna have to leave those behind as we talk about the extension axiom. As we talk about between this axioms, there is no way of having a finite geometry which satisfies the extension axiom. Axiom B3 is gonna be known as trichotomy, which says that if you have three collinear points, they're distinct. So you have ABC, these are three non-collinear, excuse me, three collinear points, then one and only one of the three between this relationships has to happen. So I want you to compare this to the first axiom. The first axiom says if you have a between this relation, then they have to be collinear. Trichotomy says that if you have three collinear points, then they're between each other and it's one or the other. You have A-B-C, that's one, or you could have A-C-B, something like that, or you have something like, so B was between A and C, C is between A or B, or you have that A is between, is between B and C, like so. And because of symmetrization, these things can be flipped around. So the idea here is that either B is between A and C, C is between A and B, or A is between B and C. One of those things happens when you have three collinear points. And then the last one, this was kind of the hardest one to understand, but it's very important here is posh's axiom. We have three distinct points, A, B, and C. If we're gonna let L be a line that's not containing A, B, and C, if L does intersect the segment A, B, then it has to intersect the segment A, C, or B, C, but it doesn't intersect both of them. So what do we mean by this? So the idea is, the way we wanna think about it is we have like a triangle, a triangle whose vertices are A, B, and C, like so, and we have a line that doesn't contain any of these three points. So it doesn't contain A, B, or C. So if our line L intersects, let's see with my notation, intersect A, B, if L intersects A, B, the segment, so it intersects some point between A and B, then it's gonna have to either intersect A, C, or it's gonna have to intersect BC, but it won't intersect both of them. So intuitively posh's axiom says if a line goes into a triangle through one side, then it must come out through another side. And so posh's axiom is very important because it allows us guaranteed intersections. So if a line intersects one line segment, then we do have some guarantee that it'll intersect another line segment. And we're gonna use posh's axiom a lot, even though it might not seem very obvious, but the idea is there's an intersection, if there's an intersection between these two points, then there's gonna be an intersection between two other points as well. And so with that, we then can define the notion of a ordered geometry. We say that an incidence geometry which satisfies the four between this axis, we just listed on the screen, that we call this an ordered geometry. The word order will make more sense in a future lecture, so we're gonna stick around with it just for right now. We get that ordered geometry. Some examples of this, Euclidean geometry R2 is an ordered geometry, but we also get that hyperbolic geometry H2 is an ordered geometry. It satisfies the incidence axis and we have this idea of the betweenness. The real projected plane is not an ordered geometry. It does satisfy the incidence axioms, but we don't have order going on here because we don't have the betweenness. So if you think of like the hemisphere model for the real projected plane, something like this, imagine we have a line, which would look something like this, you have this wraparound feature, right? Like so. What if we take three points on that line? Who is between who? It's like if we call this point A, B and C. Who's between who? Is it A dash B dash C? Well, it kind of seems like it because like, oh, I take the line segment and you can connect A to C there, but it could also be something like the following is C between A and B because after all, I could take this line segment like so and that would seem to suggest that we have B dash C dash A. So maybe C is between them, but maybe A is between them, right? A is between B and C because after all C, you go something like this and the line from B to C contained A. So maybe it's really that C dash A dash B. Which one is it? It turns out that we really can't decide in the projective model. The same problem happens in spherical geometry as well, which middle is spherical geometry is not an incidence geometry, but it doesn't satisfy the between this axioms either because trichotomy is violated in the situation axiom B3. So when we study projective in elliptic geometries in the future, we're gonna have to be more careful that while we are able to fix the incidence axioms from spherical geometry, spherical geometry didn't satisfy the elliptic parallel, excuse me, it didn't satisfy line determination, didn't satisfy elliptic parallel postulate. We could fix spherical geometry by going to the real projective plane. So now we have a projective geometry, but projective geometries, elliptic geometries are not gonna satisfy trichotomy. They have issues with between this. So that's something we have to deal with in the future. But interested enough, clearly Euclidean geometry has between this, but hyperbolic geometry does as well. And as we go into the future, we're gonna explore these topics. So as we end this video, I wanna prove one theorem about order geometry. I'm not gonna prove everything in this one video, but I wanna just give us a taste of it. So using the between this axioms, we can prove the following things. If A and B are two distinct points in order geometry, then the segment AB is equal to the intersection of the rays AB and BA, for which the order matters here. And likewise, the line determined by A and B are equal to the union of the rays AB and BA. So if we think about it intuitively as a geometry, that seems to make sense, but we can't use intuition to prove. I mean, that's to say intuition by itself is not a proof. We can help write proofs using intuition, but it has to be justified by the axioms. What we're gonna do is we're gonna prove this one together in this video, and then I'm gonna leave it up to the viewer to prove the second statement using a similar argument. So the first direction's easy. So we wanna prove that AB is equal to the intersection of the rays AB and BA. But by definition, remember, what is a ray? The ray BA, like so, we define this to be the line segment AB union some other stuff. So all these points C such that C extends the segment AB. Like so. So in particular, the ray AB contains the segment AB by definition. BA, what it's gonna look like would look like this. It's gonna have the segment BA union. We want all the points D such that B-A-D. So we extend in that manner. Now by axiom one symmetrization, the line segments BA and AB are actually the same thing. So very naturally, both rays contain the segment AB. So we get that direction first of all. So AB segment is a subset of AB-ray and the ray BA. So it'll be then a subset of their intersection like we see. So okay, so that's the first direction. We didn't even show that this is a subset of AB. So imagine we take a point that's in the intersection of the rays AB and BA. So since it's an intersection, we have that P is inside the ray AB. We also have P is inside the ray BA. But to be inside the ray AB means you're either in the segment or B is between A and P. Similar reasoning says that if P is inside the ray BA, P is either in the segment BA or A is between B and P. So clearly, if this first condition is satisfied or this condition satisfied, since AB is equal to BA, right, the line segments, if either of those conditions are satisfied, then we're in the set we wanna be in, we're done. So we're gonna assume that P satisfies this between this relationship and this between this relationship. So A-B-P and B-A-P right there. But this is a problem with trichotomy because the trichotomy actually says that B can't be between A and P and A be between B and P there. That's too many between this relationships. That's violation of trichotomy. So you can't have this situation. You can't have this and that. So that means either this happens or this happens in which case that's exactly what we want. P is inside of this and therefore since P is inside AB, P was an arbitrary point, we have that as a subset. And that then proves this equality right here. And like I said, I will leave it to the viewer to prove this equality as an exercise of order geometry.